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Spectral signal-to-noise ratio and resolution assessment of 3D reconstructions M. Unser a , C.O.S. Sorzano a,b,c, * , P. The ´venaz a , S. Jonic ´ a , C. El-Bez d , S. De Carlo e , J.F. Conway f , B.L. Trus g a Biomedical Imaging Group, Swiss Federal Institute of Technology Lausanne, CH-1015 Lausanne VD, Switzerland b Escuela Polite ´cnica Superior, Universidad San Pablo-CEU, Campus Urb. Monteprı ´ncipe s/n, 28668 Boadilla del Monte, Madrid, Spain c Biocomputing Unit, National Center of Biotechnology (CSIC), Campus Univ. Auto ´ noma s/n, 28047 Cantoblanco, Madrid, Spain d Laboratoire dÕanalyse ultrastructurale, Universite ´ de Lausanne, CH-1015 Lausanne VD, Switzerland e Department of Molecular and Cell Biology, Howard Hughes Medical Institute, University of California, Berkeley, CA 94720, USA f Laboratoire de Microscopie Electronique Structurale, Institut de Biologie Structurale, 41 rue Jules Horowitz, 38027 Grenoble, Cedex 1, France g Imaging Sciences Laboratory, Center of Information Technology (NIH/DHHS), 12 Center Drive, MSC 5624, Bethesda, MD 20892-5624, USA Received 19 January 2004, and in revised form 22 September 2004 Available online 29 December 2004 Abstract Measuring the quality of three-dimensional (3D) reconstructed biological macromolecules by transmission electron microscopy is still an open problem. In this article, we extend the applicability of the spectral signal-to-noise ratio (SSNR) to the evaluation of 3D volumes reconstructed with any reconstruction algorithm. The basis of the method is to measure the consistency between the data and a corresponding set of reprojections computed for the reconstructed 3D map. The idiosyncrasies of the reconstruction algorithm are taken explicitly into account by performing a noise-only reconstruction. This results in the definition of a 3D SSNR which pro- vides an objective indicator of the quality of the 3D reconstruction. Furthermore, the information to build the SSNR can be used to produce a volumetric SSNR (VSSNR). Our method overcomes the need to divide the data set in two. It also provides a direct mea- sure of the performance of the reconstruction algorithm itself; this latter information is typically not available with the standard resolution methods which are primarily focused on reproducibility alone. Ó 2004 Elsevier Inc. All rights reserved. 1. Introduction The three-dimensional (3D) reconstruction of biolog- ical macromolecules provides structural biologists with key information to fully understand the properties and functions of a given complex. 3D Electron Microscopy (3DEM) of single particles is one of the most useful imaging techniques since it allows the visualization of biological macromolecules nearly in their native state without any constraint of size or any need to crystallize (Baumeister and Steven, 2000; Ellis and Hebert, 2001; Frank, 2002; Ruprecht and Nield, 2001; Sali et al., 2003). The physical limits of 3DEM resolution have been discussed in terms of macromolecule size, micro- scope features, and number of images (Henderson, 1995). However, the assessment of the quality actually achieved by a 3D reconstruction is still an open problem (Grigorieff, 2000). To the best of our knowledge, the measures available to assess the quality of 3D reconstructions in single-par- ticle electron microscopy are: the Fourier shell correla- tion (FSC) (Harauz and van Heel, 1986; Saxton and Baumeister, 1982; van Heel, 1987), the Fourier ring phase residual (FRPR) (van Heel, 1987), the Q factor (Kessel et al., 1985; van Heel, 1980), and the spectral sig- nal-to-noise ratio (SSNR) (Penczek, 2002; Unser et al., 1996). 1047-8477/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jsb.2004.10.011 * Corresponding author. Fax: +34 91 585 4506. E-mail address: [email protected] (C.O.S. Sorzano). www.elsevier.com/locate/yjsbi Journal of Structural Biology 149 (2005) 243–255 Journal of Structural Biology

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Journal of

www.elsevier.com/locate/yjsbi

Journal of Structural Biology 149 (2005) 243–255

StructuralBiology

Spectral signal-to-noise ratio and resolution assessmentof 3D reconstructions

M. Unsera, C.O.S. Sorzanoa,b,c,*, P. Thevenaza, S. Jonica, C. El-Bezd, S. De Carloe,J.F. Conwayf, B.L. Trusg

a Biomedical Imaging Group, Swiss Federal Institute of Technology Lausanne, CH-1015 Lausanne VD, Switzerlandb Escuela Politecnica Superior, Universidad San Pablo-CEU, Campus Urb. Monteprıncipe s/n, 28668 Boadilla del Monte, Madrid, Spain

c Biocomputing Unit, National Center of Biotechnology (CSIC), Campus Univ. Autonoma s/n, 28047 Cantoblanco, Madrid, Spaind Laboratoire d�analyse ultrastructurale, Universite de Lausanne, CH-1015 Lausanne VD, Switzerland

e Department of Molecular and Cell Biology, Howard Hughes Medical Institute, University of California, Berkeley, CA 94720, USAf Laboratoire de Microscopie Electronique Structurale, Institut de Biologie Structurale, 41 rue Jules Horowitz, 38027 Grenoble, Cedex 1, Franceg Imaging Sciences Laboratory, Center of Information Technology (NIH/DHHS), 12 Center Drive, MSC 5624, Bethesda, MD 20892-5624, USA

Received 19 January 2004, and in revised form 22 September 2004Available online 29 December 2004

Abstract

Measuring the quality of three-dimensional (3D) reconstructed biological macromolecules by transmission electron microscopy isstill an open problem. In this article, we extend the applicability of the spectral signal-to-noise ratio (SSNR) to the evaluation of 3Dvolumes reconstructed with any reconstruction algorithm. The basis of the method is to measure the consistency between the dataand a corresponding set of reprojections computed for the reconstructed 3D map. The idiosyncrasies of the reconstruction algorithmare taken explicitly into account by performing a noise-only reconstruction. This results in the definition of a 3D SSNR which pro-vides an objective indicator of the quality of the 3D reconstruction. Furthermore, the information to build the SSNR can be used toproduce a volumetric SSNR (VSSNR). Our method overcomes the need to divide the data set in two. It also provides a direct mea-sure of the performance of the reconstruction algorithm itself; this latter information is typically not available with the standardresolution methods which are primarily focused on reproducibility alone.� 2004 Elsevier Inc. All rights reserved.

1. Introduction

The three-dimensional (3D) reconstruction of biolog-ical macromolecules provides structural biologists withkey information to fully understand the properties andfunctions of a given complex. 3D Electron Microscopy(3DEM) of single particles is one of the most usefulimaging techniques since it allows the visualization ofbiological macromolecules nearly in their native statewithout any constraint of size or any need to crystallize(Baumeister and Steven, 2000; Ellis and Hebert, 2001;Frank, 2002; Ruprecht and Nield, 2001; Sali et al.,

1047-8477/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.jsb.2004.10.011

* Corresponding author. Fax: +34 91 585 4506.E-mail address: [email protected] (C.O.S. Sorzano).

2003). The physical limits of 3DEM resolution havebeen discussed in terms of macromolecule size, micro-scope features, and number of images (Henderson,1995). However, the assessment of the quality actuallyachieved by a 3D reconstruction is still an open problem(Grigorieff, 2000).

To the best of our knowledge, the measures availableto assess the quality of 3D reconstructions in single-par-ticle electron microscopy are: the Fourier shell correla-tion (FSC) (Harauz and van Heel, 1986; Saxton andBaumeister, 1982; van Heel, 1987), the Fourier ringphase residual (FRPR) (van Heel, 1987), the Q factor(Kessel et al., 1985; van Heel, 1980), and the spectral sig-nal-to-noise ratio (SSNR) (Penczek, 2002; Unser et al.,1996).

244 M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255

By far, the most frequently used method to measurereconstruction quality is the FSC. However, the thresh-old at which the FSC defines the reconstruction resolu-tion is still a questionable topic. Some classicalthresholds are 0.8, 0.67, 0.5, 0.3, or 2 or 3 times the cor-relation with random noise (Penzcek, 1998). More re-cently, the new threshold 0.143 has appeared(Rosenthal and Henderson, 2003).

The main drawback of the FSC, as well as of theFRPR, is that they determine the 3D resolution by firstcomputing two independent reconstructions and then bytesting the concordance of their 3D Fourier transforms.This approach provides a fair assessment for the repro-ducibility of the experiment, but has also some basic lim-itations. First, it constrains us to producereconstructions using subsets of the available data. Sec-ond, it does not explicitly test for the validity of thereconstruction algorithm itself—reproducibility aloneis not a guarantee that the calculated density accuratelyrepresents the raw data. These problems are avoided inthe method developed by Conway et al. (1993) whichperforms a FSC resolution assessment in the data do-main using a reprojected version of the 3D reconstruc-tion. This measure is also known in the literature asFRC3D. However, this non-standard way of using theFSC raises the difficulty of determining a resolutionthreshold that is sound, mainly because of the very dif-ferent statistical properties of the images being com-pared (i.e., data versus reprojection).

Another more serious limitation of using FSC is thatthis correlation-based measure turns out to be invariantto any isotropic filtering of the entire data set. This issimply because such a global filtering will manifest itselfby a radial rescaling of all spectral components both onthe input (data) and on the output (3D reconstructionand reprojections). If the scaling is isotropic, the propor-tionality factor is the same for all spectral componentsat a given radial frequency with the result that the cor-responding FSC value remains unchanged. Thus, wemay very well perform lowpass filtering of the wholedata set and still obtain the same FSC resolution esti-mate as before, which is obviously not satisfactory.

This motivated us to propose (Unser et al., 1996) analternative criterion which is inspired from previouswork in 2D (Unser et al., 1987). This criterion takes intoaccount the whole chain of events in the reconstructionprocess and has a simple intuitive interpretation. Thisidea was further developed by Penczek (2002). However,this latter work is restricted to the class of algorithmsthat performs the reconstruction by interpolation inthe Fourier space. In this paper, we extend the 3D useof the SSNR independently of the reconstruction algo-rithm. Computer simulations in a well-controlled envi-ronment, as well as results with real data, point outthe validity of our extension. The image processingpackage Xmipp (Marabini et al., 1996) incorporates

the resolution measure defined in this paper. The pack-age can be found at http://www.cnb.uam.es/~bioinfo.

2. SSNR resolution assessment

2.1. Basic assumptions

Our data set X ¼ fxðiÞk;l; i ¼ 1; . . . ; Ig consists of I inde-pendent projective views xðiÞk;l, where the spatial locationis indexed by (k, l). We make the standard assumptionof an additive signal + noise model: X ¼ SþN, whereS and N denote the signal and noise components,respectively. Note that this assumption can also be trea-ted as a definition: we may define the signal to be the ex-plained portion of the data which is common to allmeasurements, and the noise to be the unexplained part(residue). Ideally, we would like the noise component tobe due to random fluctuations only, but it can poten-tially also account for a whole variety of perturbing fac-tors such as the presence of outliers and/or an incorrectestimation of the imaging geometry.

The second assumption is that the reconstruction pro-cess is linear in the sense that RecfXg ¼ RecfSgþRecfNg (RecfXg is the volume reconstructed from thedata set X). We only require that this relation holds inthe case when all the reconstruction parameters are thesame. This is a quite reasonable hypothesis becausethe ray transform is a linear operator (Natterer andWubbeling, 2001). Thus, any reasonable reconstructionalgorithm that attempts to invert the ray operator shouldbe linear as well.

2.2. Spectral signal-to-noise ratio

Our method estimates the relative energy contribu-tion of the reconstructed signal and noise components(RecfSg and RecfNg) by performing two independentcomputations. The first assesses the consistency betweenthe input data and the 3D map produced by the tomo-graphic reconstruction algorithm. The second dealsexclusively with the effect of the algorithm on the noisecomponent N. Both types of estimates are combinedinto a global SSNR which characterizes the overallbehavior of the system for the particular data set athand.

2.2.1. Data consistency

Let X ðiÞK;L, denote the 2D discrete Fourier transform of

the input image xðiÞk;l. By convention, we use (K,L) as thespatial frequency indices. After determination of the rel-ative orientations of the individual views, the tomo-graphic reconstruction algorithm produces a 3D mapof the underlying specimen. This 3D map (or model) isthen used to generate a corresponding set of reprojectedimages f~xðiÞk;l; i ¼ 1; . . . ; Ig with Fourier transforms

M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255 245

feX ðiÞK;L; i ¼ 1; . . . ; Ig; these provide an estimate of the sig-

nal that is initially present in our data. By considering aregion RDx (x) in Fourier space that corresponds to anannulus with central radial frequency x and width Dx,we then calculate the input SSNR as

ISSNRðX;x;DxÞ ¼PI

i¼1

PK;L2RDxðxÞ

eX ðiÞK;L

��� ���2PIi¼1

PK;L2RDxðxÞ X

ðiÞK;L � eX ðiÞ

K;L

��� ���2 :ð1Þ

The numerator in (1) is an estimate of the signalpower centered at frequency x, while the denominatoris a measure of the corresponding input noise energy(unexplained portion of the data). Thus, the ratio ofboth quantities yields an estimate of the SNR at the in-put of the system, that is, the SNR prior to 3Dreconstruction.

2.2.2. Noise reduction factor

When I images are combined to yield a 3D recon-struction, there is some form of averaging involvedwhich causes the noise to be reduced. To characterizethis effect, we introduce the noise reduction factor ofthe algorithm, a (x,Dx), which is typically not constant.The major difficulty here is that a (x,Dx) depends onmany application-specific parameters; for example, thetype of tomographic algorithm used, the imagingparameters (angles), the type of symmetry (e.g., icosahe-dral), and the number of views. It is therefore very diffi-cult in general to determine a (x,Dx) analytically. Oursolution is to estimate a (x,Dx) empirically by injectingwhite Gaussian noise into the reconstruction algorithmwith all parameters being the same as for X. Practically,this is equivalent to using the estimate

aðx;DxÞ ¼PI

i¼1

PK;L2RDxðxÞ

eN ðiÞK;L

��� ���2PIi¼1

PK;L2RDxðxÞ N

ðiÞK;L

��� ���2 ; ð2Þ

where N now represents noise-only images, and whereeN denotes the corresponding reprojection calculatedfrom the noise-only 3D reconstruction map. It is impor-tant to use white noise at this step since the goal is tomeasure the attenuation introduced by the reconstruc-tion algorithm. Therefore, all frequencies must be pres-ent in the same amount at the input of the system. Notethat we have aðx;DxÞ � ISSNRðN;x;DxÞ, except thatthere is no subtraction in the denominator because theunderlying signal is zero by definition.

Unser et al. (1987) showed that the 2D SSNR in theabsence of signal, which is the 2D analog of what in thispaper is called a (x,Dx), follows a central F-Snedecordistribution with r1 = nR and r2 = (I�1)nR degrees offreedom. There, nR is the number of elements participat-ing in the annulus RDx (x). The variance of such a distri-bution is ð r2

r2�2Þ2 2ðr1þr2�2Þ

r1ðr2�4Þ (Rade and Westergeren, 1999).

Note that this variance tends to 2/r1 when the number ofimages tends to infinity, and to 0 when the number ofpoints in the annulus tends to infinity. The statisticalassumptions made for the derivation of this distributionare no longer valid in the 3D case and finding an analyt-ical statistical distribution remains an open problem.However, these two cases are conceptually similar andone should still expect a smaller variance of a (x,Dx)as the number of images or the annulus size increases.

2.2.3. SSNR estimate

Finally, we combine Eqs. (1) and (2) to obtain a mea-sure of the true SSNR on the reconstructed signal

SSNRðX;x;DxÞ ¼ max 0;ISSNRðX;x;DxÞ

aðx;DxÞ � 1

� �:

ð3ÞThe main reason for subtracting 1 from the ratio is to

produce an unbiased estimate. In particular, we wantthe SSNR to be zero when the data consists of noiseonly (i.e., X ¼ N). Note that (3) is the 3D extension ofthe 2D SSNR criterion for correlation averaging (Unseret al., 1987). In this former simpler case, a (x,Dx) can bedetermined analytically; it is simply 1

I�1for any (x,Dx).

As in this previous work, an operational resolutionlimit can be specified as the spatial frequency at whichthe SSNR falls below an acceptable baseline. Alterna-tively, we may also assess the quality of our 3D recon-struction by comparing the ISSNR curves for the twomodalities: X (data = signal + noise) and N (noiseonly).

2.3. Volumetric spectral signal-to-noise ratio

The 3D distribution of the SNR in the frequencyspace is strongly dependent on the angular distributionand might not be radially symmetric. An even angulardistribution is a sufficient condition to guarantee a fullcoverage of the 3D Fourier space, although this condi-tion is not necessary (Orlov, 1976). On the other ex-treme, experimental conditions might force an angulardistribution with a strongly inhomogeneous frequencycoverage. For instance, in random conical tilt (Raderm-acher, 1988), there is a region in Fourier space (missingcone) where no information is available. However,although in this region the SNR is zero, it is taken intoaccount to compute the radial average implied byRDx(x).

Other parameters control the asymmetry of the SNR,like the number of projections from each direction. If acertain projection direction is more populated than oth-ers, then the resolution in this direction should be largersince more information is available.

The previous SSNR formulas can be applied to per-form individual SSNR estimations specific to each pro-jection image by simply setting RDx (x) = x, instead of

246 M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255

an annulus as was done in the previous section. In thisway for each input projection x(i), we can associate anindividual SSNR(i) defined as

SSNRðiÞK;L ¼ max 0;

ISSNRðiÞK;L

aðiÞK;L� 1

( ); ð4Þ

where

ISSNRðiÞK;L ¼

eX ðiÞK;L

��� ���2X ðiÞ

K;L � eX ðiÞK;L

��� ���2 ;

aðiÞK;L ¼eN ðiÞ

K;L

��� ���2N ðiÞ

K;L

��� ���2 :Note that SSNR(i) is a real-valued image defined in

Fourier space. The central-slice theorem (Natterer andWubbeling, 2001) states that the 2D Fourier transformof a ray projection of a volume is one of the slices ofthe 3D Fourier transform of that volume. We makeuse of this theorem to fit the set of images {SSNR(i);i = 1, . . .,N} by a real-valued volume in Fourier space,VSSNR(x). This volume provides an estimate of theSNR frequency distribution.

For the volumetric interpolation in Fourier space, weapproximate the volume VSSNR by a finite seriesexpansion of the form

VSSNRðxÞ �XJ

j¼1

cjbðx� xjÞ; ð5Þ

where x is the frequency at which this volume is approx-imated, xj are fixed points distributed in a homogeneousgrid, b is a basis function, and the cjs are the coefficientsof the expansion. Following Matej and Lewitt (1995)and Matej and Lewitt (1996), we use generalized Kai-ser–Bessel window functions as the basis function b dis-tributed over a body-centered cubic grid (BCC). Criteriafor the selection of the blob parameters are given in theAppendix A.

We find VSSNR(x) as the solution of the equationsystem

VSSNR xðiÞK;L

� �¼ SSNRðiÞ

K;L; ð6Þ

where xðiÞK;L is the frequency in the 3D Fourier space of

the sample (K,L) of the image SSNR(i). This frequencyis given by the central-slice theorem.

Due to the huge dimensions of this equation system,we solve it in an iterative fashion using the Block Alge-braic Reconstruction Technique (Herman, 1998), whereeach block is defined by the equations corresponding tothe same image SSNR(i).

Penczek (2002) also proposed the use of this volumet-ric SSNR although its computation was restricted to the

class of tomographic algorithms that reconstruct by ex-plicit interpolation in Fourier space. In that work it wasshown that anisotropic low-pass filters could be specifi-cally tailored to the spectral distribution of the informa-tion. It was also proposed to use the 3D inertia matrix todetermine the directions of maximal and minimalamount of information. These techniques are still appli-cable to our VSSNR(x) volume.

3. Results

Several experiments were carried out to check thevalidity of our resolutionmeasure. Computer simulationsin a well-controlled environment were performed to showthe applicability of the SSNR in relevant situations. Then,the SSNR was tested on experimental electron-micros-copy data. In particular, we used the cryo-negative stain-ing data of GroEL obtained by De Carlo et al. (2002)versus reprojection). This particle was selected becauseits atomic model is available; therefore, the quality ofthe reconstruction can also be established objectively.

3.1. Computer-simulated experiments

Computer-simulated experiments were carried outaccording to the following scheme: starting from a knownvolume called phantom, projection images are simulated.These images are input to a tomographic algorithm pro-ducing a 3D reconstruction. The quality of this recon-struction is assessed by comparison with the phantom.

The Halobacterium halobium bacteriorhodopsin tri-mer was taken as phantom from the MacromolecularStructure Database (Protein Quaternary Structurequery, PQS, Boutselakis et al., 2003) (PQS entry:1BRD, Henderson et al., 1986). ART + blobs (Marabiniet al., 1998) was used as reconstruction algorithm. Thethreefold symmetry of the bacteriorhodopsin was notexplicitly used during the reconstruction.

The computation of the FSC is performed by directcomparison of the phantom with the reconstruction.Since the ideal volume is available, this comparison isno longer a reproducibility measure but a true resolutionestimation. From now on, we will refer to this measureas Reference FSC (FSCref). Furthermore, there is noneed to split the experimental image set in two halves.The FSC between two independent reconstructions is re-ferred to in this paper as FSC.

The sampling rate in Fourier space for all the simu-lated experiments was 0.005 A�1. The width Dx of theannulus RDx (x) was taken as 0.020 A�1.

3.1.1. Resolution estimation

In this experiment, we test the ability of SSNR toestablish the resolution of a reconstruction. A nearlyeven angular distribution with 1000 images (see Fig. 1)

Fig. 1. Nearly even angular distribution. A small triangle is placed atthe tip of the unit vectors representing each projection direction. Thein-plane rotation angle is represented by the relative rotation of eachtriangle. The separation between projections for this distribution isapproximately 6�.

Fig. 2. SSNR (top), FSC, and FSCref (bottom) of the reconstruction ofthe bacteriorhodopsin from simulated data with an even angulardistribution. The SSNR is also shown in logarithmic scale (dB)according to the formula SSNR(dB) = 10log10 (SSNR).

M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255 247

was simulated. White Gaussian noise was added to theideal projection images to achieve an average SNR ofapproximately 1/3. The corresponding SSNR, FSC,and FSCref are represented in Fig. 2. Note that the mag-nitude of the FSC is substantially below that of theFSCref. The resolution at SNR = 1 as computed bySSNR is 1/25 A�1. This frequency implies a thresholdof 0.75 in the FSCref and 0.38 in the FSC. Commonthresholds for the FSC are 0.5 which is known to beconservative, and 0.3 which is currently the most used.The resolution at a FSC threshold of 0.5 is 1/31 A�1

and at a threshold of 0.3 is 1/22 A�1. These resolutionsare achieved by the SSNR curve at thresholds 1.82 (>1)and 0.63 (<1). These SSNR thresholds are in agreementwith the fact that the FSC threshold 0.5 is conservative,although 0.3 seems to be a little bit optimistic.

We performed the same reconstruction usingWeighted Back-Projection [WBP, Radermacher(1992)]. The estimated resolution using SSNR atSNR = 1 was 1/25 A�1. At this frequency, the valuesof FSCref and FSC were 0.81 and 0.42, respectively.The resolution estimate for the cutoff value FSC = 0.5was 1/28.6 A�1. The threshold limit FSCref = 0.75 yieldsthe value 1/22 A�1. In both cases, the resolutions esti-mated by the SSNR and the FSC seem to be an under-estimation of the resolution computed by the FSCref.

3.1.2. Variability of SSNR

Our resolution measure uses an estimate of the atten-uation factor at each frequency. This attenuation isestablished by injecting white Gaussian noise into the

reconstruction algorithm while keeping the remainingparameters (angular distribution, reconstruction freeparameters, . . .) the same. It is interesting to knowwhether this attenuation is reliably estimated. In thisexperiment, we compute the attenuation factora (x,Dx) for the previous experiment after injecting dif-ferent noise realizations. Fig. 3 shows the plot ofa (x,Dx) versus frequency for 10 different noiserealizations.

3.1.3. Simulations with geometrical errors

Although image noise is one of the error sources insingle particle EM, it is not the most limiting factor sinceits effect can be easily removed by the incorporation ofmore images into the reconstruction process. Geometri-cal errors coming from uncertainties in the projectiondirection and position of the image center severely affectthe achievable resolution (Penczek et al., 1994). In thisexperiment, projection images are computed accordingto the even angular distribution shown in Fig. 1. How-ever, we supply the reconstruction algorithm with noisyangular information. In particular, we add Gaussian

Fig. 3. Different realizations of the attenuation factor for the angulardistribution of Fig. 1.

Fig. 5. SSNR (top), FSC, and FSCref (bottom) of the reconstruction ofthe bacteriorhodopsin from simulated data with a perturbed angulardistribution. The SSNR is also shown in logarithmic scale (dB)

248 M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255

noise with zero mean and a standard deviation of 5� tothe Euler angles defining the projection directions. Theperturbed angular distribution is shown in Fig. 4. Theparticle origin is also randomly shifted horizontallyand vertically. The shift in both directions follows a nor-mal distribution with zero mean and a standard devia-tion of 6.54 A. White Gaussian noise was added to theideal projection images to achieve an average SNR ofapproximately 1/3. The corresponding SSNR, FSC,and FSCref are shown in Fig. 5. The resolution atSNR = 1 as computed by SSNR is 1/42 A�1. The refer-ence resolution at FSCref = 0.75 is 1/38 A�1. Finally, theresolution at FSC = 0.5 is 1/38 A�1 and at FSC = 0.3 is1/34 A�1.

Fig. 4. Even angular distribution perturbed by angular noise. Arandom number (normally distributed with zero mean and standarddeviation 5�) is added to each of the three Euler angles describing theprojection directions in Fig. 1.

according to the formula SSNR(dB) = 10log10 (SSNR).

3.1.4. Simulations with CTF

EM images are subjected to microscope aberrationsthat also affect the maximum resolution achievable(Frank, 2002). In this experiment, we consider the effectof the contrast transfer function (CTF) of the micro-scope. We have simulated a CTF with the followingparameters: accelerating voltage = 200 kV, defo-cus = �2.8 lm, spherical aberration = 2 mm, conver-gence cone = 0.21 mrad (Velazquez-Muriel et al., 2003)(see Fig. 6). The angular distribution is even (see Fig.1) and a SNR of 1/3 was simulated. The correspondingSSNR, FSC, and FSCref are shown in Fig. 7. The reso-lution at SNR = 1 as computed by SSNR is 1/31 A�1.The reference resolution at FSCref = 0.75 is 1/30 A�1.Finally, the resolution at FSC = 0.5 is 1/30 A�1 and atFSC = 0.3 is 1/29 A�1. The first zero of the CTF is at1/27 A�1.

3.1.5. Volumetric SSNR

To check the usefulness of the volumetric SSNR,three experiments were carried out. They all simulatethe CTF previously described and the SNR is 1/3. Three

Fig. 6. Radial profile of the contrast transfer function used forcomputer simulations.

Fig. 7. SSNR (top), FSC, and FSCref (bottom) of the reconstruction ofthe bacteriorhodopsin from simulated data with an even angulardistribution when the microscope aberrations are simulated. TheSSNR is also shown in logarithmic scale (dB) according to the formulaSSNR(dB) = 10log10 (SSNR).

Fig. 8. Uneven angular distribution. Projections are randomly dis-tributed on the projection space although top views are more frequentthan lateral ones.

Fig. 9. Uneven distribution with a missing cone. Projections aredistributed randomly on the projection space. Top views are morefrequent and no projection is taken with a tilt angle greater than 45�.

M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255 249

different angular distributions were used: even (Fig. 1),uneven (Fig. 8), and uneven distribution with a missingcone (Fig. 9). Each distribution has 1000 different pro-jection directions. The blobs and grid used for the volu-metric interpolation were those referred to as ‘‘standard

blob’’ in Matej and Lewitt (1996). Its interpolatingproperties are described by Matej and Lewitt (1996)and Garduno and Herman (2001). The Appendix pro-vides some guidelines for its use. Isosurfaces of theresulting volumetric SSNRs are shown in Fig. 10. Twoisosurfaces are shown on each graph, one correspondingto SNR = 1 and another to SNR = 4. Notice the rela-tionship between the SSNR isosurface shape and theangular distribution: the even distribution has a nearlyisotropic SSNR (i.e., the resolution achieved in eachdirection is approximately the same), the uneven distri-bution shows larger SSNR in the plane perpendicularto the projection direction more populated (the resolu-tion in the directions lying in the horizontal plane are

Fig. 10. Top and side view of the volumetric SSNR for the reconstruction of bacteriorhodopsin from simulated data using an even (top), an uneven(middle), and an uneven distribution with a missing cone (bottom). The mesh corresponds to the isosurface of SSNR = 1. The solid isosurfacecorresponds to a SSNR = 4.

250 M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255

larger than in other directions), and the uneven distribu-tion with a missing cone shows a region of missing infor-mation aligned with the missing cone (very littleresolution is achieved in those directions within themissing cone).

3.1.6. Tomographic experimentAn interesting question is whether this methodology

can be applied to the reconstruction of a few projec-tions. The question is important because it refers to elec-tron tomography of non-repeatable structures like cells.

M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255 251

To test this end, we simulated a cross-section slice of anorganelle with small proteins inside in arbitrary posi-tions and orientations (see Fig. 11). One hundred andtwenty images were taken in a single-axis tilt series withtilt angles evenly distributed between �60� and 60�. Twoiterations (over the whole data set) of ART + blobswere employed for this reconstruction with a relaxationfactor of 0.2 and 0.1, respectively. The correspondingVSSNR is shown in Fig. 12. The missing wedge can beseen in the top view. A striking feature is that the reso-lution is strongly anisotropic. It has much higher resolu-tion in the direction of the tilt axis than elsewhere. This

Fig. 11. (Top) Isosurface of a phantom simulating a cross-section ofan organelle with a set of proteins at random orientations. (Bottom)Tomographic reconstruction of the phantom at two different orienta-tions. On the right image it can be seen that the organelle wall is verywell defined along the tilt axis (vertical axis of this image) while there isa huge uncertainty along the perpendicular direction (horizontal axisof this image).

Fig. 12. Top and side view of the volumetric SSNR for thetomographic reconstructions. The isosurface correspond to aSSNR = 1.

is because the reconstructed volume shows a high levelof details at the tilt axis (notice the sharp transition atthe organelle wall). In fact, the sinc-like envelope ob-served in the side view stems from the organelle wall.The resolution in a direction perpendicular to the tiltaxis is about 209 A, while the resolution along the tiltaxis can be as high as 7 A (the sampling rate was setto 3.27 A). Of course, this high resolution is onlyachieved in the tilt axis. This points out a disadvantageof the VSSNR methodology when applied to electrontomography: it does not show the local resolution. Infact, this is a drawback of any Fourier-based measurebecause of the tradeoff between spatial and frequencyresolution (Mallat, 1999).

3.2. Results on experimental data

To test the applicability of the proposed resolutionmeasure to experimental electron-microscopy data, thecryo-negative micrographs of GroEL taken by (De Car-lo et al., 2002) were used. 2610 projection images wereinvolved. The GroEL has two rotational symmetry axis:one of order 7 around the vertical axis and another oneof order 2 around a lateral axis. This implies that everyprojection image is identical to other 13 (7 · 2 � 1)views. Symmetry was explicitly taken into account dur-ing the reconstruction process; thus, 36540 (2610 · 14)images were used. The reconstruction was performedwith ART + blobs (Marabini et al., 1998) and was basedon an angular assignment (Penczek, 2002) with a sam-pling step of 3�. The X-ray model of GroEL availablein PDB [Berman et al. (2000), PDB entry: 1GRL, Braiget al. (1994, 1995)] was used as phantom for the compu-tation of the FSCref. Fig. 13 shows the correspondingSSNR, the FSC, and FSCref. The resolution atSNR = 1 as computed by SSNR is 1/26 A�1, while theresolution at FSCref = 0.75 is 1/36 A�1. The resolutionat a threshold of FSC = 0.5 is 1/22 A�1 and atFSC = 0.3 is 1/17 A�1. (The sampling rate in Fourierspace is 0.004 A�1 and the width Dx of the annulusRDx(x) was taken as 0.016 A�1.) Finally, the volumetricSSNR was computed using the same blobs as for thecomputer-simulated experiments. The VSSNR is shownin Fig. 14. The non-uniform distribution of the volumet-ric SSNR can be explained by the uneven distribution ofthe tilt angle, see Fig. 15.

4. Discussion

One of the most distinctive features of the proposedmethod of resolution assessment is that it explicitly takesinto account the noise-reduction effect of the reconstruc-tion algorithm. A very natural temptation would be toanalytically derive such noise statistics from a mathe-matical description of the algorithm. Unfortunately,

Fig. 13. SSNR (top), FSC, and FSCref (bottom) of the reconstruction ofGroEL using experimental cryo-microscopy data when compared withtheX-raymodel ofGroEL. The SSNR is also shown in logarithmic scale(dB) according to the formula SSNR(dB) = 10log10 (SSNR).

Fig. 14. Top and side view of the volumetric SSNR for thereconstruction of GroEL using experimental cryo-microscopy data.The mesh corresponds to the isosurface of SSNR = 1. The solidisosurface corresponds to a SSNR = 4.

Fig. 15. Histogram of the tilt angle of the 2160 GroEL experimentalimages.

252 M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255

the chances of success of a deductive approach are extre-mely slim because there are far too many parameters in-volved. In addition, it is often very difficult for the enduser to have a precise description of the algorithm thathe/she is using. What we propose instead, is to performa noise-only experiment to measure the performance of

the algorithm in a given configuration. This is perhapsthe simplest and most universal solution to the problem.It is applicable to any situation without requiring anyknowledge of the inner working of the algorithm whichis considered as a black box and which is assumed to belinear. Its only drawback is that it requires some morecomputer runs.

The basis for this method is to check for the consis-tency between the reconstructed map and the inputdata (note that for doing this the volume gray valuesmust be such that the reprojected images have gray val-ues in the same range as the experimental images). Inthis sense, the proposed criterion goes beyond the stan-dard reproducibility tests which are commonly used inthe field. It provides an objective assessment of thequality of the reconstruction algorithm itself by lookingat the consistency between the result and the inputdata. Most iterative reconstruction methods are basedon a similar consistency principle. Because the underly-ing problem is linear, they typically iterate by refiningresidues until convergence to a solution that minimizesthe difference between the reprojected map and theinput data.

The SSNR has a simple intuitive interpretation. Itleads to a very natural threshold-based definition ofthe resolution limit. SSNR also provides us with a finecharacterization of the quality of the reconstruction asa function of the radial frequency. The bottom line isthat we will only trust those signal frequency compo-nents whose energy is above what would have been ob-tained if the algorithm was applied to noise only.

The SSNR allows one to use the full set of images toperform the reconstruction; there is no need to dividethe data into two subsets with the subsequent lost ofresolution.

Even though the computation is performed in 2D,the SSNR is easily interpreted in 3D as shown bythe volumetric SSNR. In this extension to 3D, the

M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255 253

anisotropy (dependency with the direction) of theangular distribution is explicitly taken into accountso that projection directions with more views have ahigher noise attenuation factor. In this way, the infor-mation provided by VSSNR volume is very helpful tounderstand the relationship between the angular distri-bution and the resolution achieved in each direction.(Some directions might have been better representedor completely ignored in the projection set.) This re-sults in an uneven SSNR distribution that is easily de-tected by our measure. This anisotropy can be laterused to build tailored lowpass filters as proposed byPenczek (2002).

The application of the VSSNR methodology to elec-tron tomography data shows that it is possible to definea directional resolution measure for this kind of data.We found the VSSNR (and hence the resolution) to bemaximal in the direction of the tilt axis, which is consis-tent with what our expectations. In principle, one wouldalso expect the resolution along this axis to be the high-est in the central region where the projection rays are thedensest. Unfortunately, this cannot be assessed from thepresent spectral analysis since the Fourier transform hasno spatial localization at all. Finding a way to estimatethe resolution locally would be extremely valuable forelectron tomography. This is an important open prob-lem which will undoubtedly require some kind of com-promise because of the fundamental limits imposed bythe uncertainty principle.

In terms of computational effort, the computation ofVSSNR amounts for two extra reconstructions. First, areconstruction with pure noise under the very same con-ditions as the data must be performed. Then, the recon-struction from the experimental data and the noisyreconstruction are input into Eq. (4) that needs to per-form one projection of each volume for each experimen-tal image at hand. Finally, the interpolation indicated inEq. (6) needs be performed.

The experiments showed that the resolution esti-mates obtained by SSNR are usually quite comparableto those obtained by FSCref with a threshold of 0.75and a FSC threshold between 0.5 and 0.3 (the corre-sponding estimates are typically within a distance of 1Fourier sample.) However, when the FSC is used as aconsistency measure, very high FSC values can be ex-pected as the number of projections taken into accountgrows (Grigorieff, 2000). The interpretation of thesehigh values in terms of resolution is more difficult sincestandard thresholding rules do not hold and their appli-cation can be certainly misleading as shown in theexperiment of Section 3.1. Furthermore, a priori knowl-edge such as molecule surface, mass, or symmetry, canbe explicitly taken into account in current tomographicalgorithms (Sorzano et al., 2002). This a priori informa-tion highly increases the consistency between two inde-pendent reconstructions. These two facts make it

difficult to select a FSC threshold that defines the vol-ume resolution. However, SSNR still maintains a clearmeaning and the threshold can still be stated in terms ofthe desired SNR.

The discrepancy between the resolution for GroELestimated by SSNR (1/26 A�1) or FSC (1/22 A�1) andthe FSCref (1/36 A

�1) may be caused by the disagree-ment between the particle conformation when it is crys-tallized and when it is studied as a single particle. In anycase, the previously reported resolution (1/14 A�1)seems to be an overestimation; although the work�smain purpose was not the achievement of the highestresolution of GroEL in solution (De Carlo et al.,2002).

5. Conclusions

We extended the use of the 3D SSNR in single-par-ticle electron microscopy. This measure was alreadyintroduced in 3D by Unser et al. (1996) and was fur-ther developed by Penczek (2002). We have generalizedthe class of tomographic algorithms to which it can beapplied by following a black-box approach. This gen-eralization easily allows the estimation of the fre-quency distribution of the signal-to-noise ratio, apiece of information that is very useful to structuralbiologists.

Acknowledgments

Partial support is acknowledged to the ‘‘Comunidadde Madrid’’ through project CAM-07B-0032-2002, the‘‘Comision Interministerial de Ciencia y Tecnologıa’’of Spain through projects BIO2001-1237, BIO2001-4253-E, BIO2001-4339-E, BIO2002-10855-E, the Euro-pean Union through Grants QLK2-2000-00634,QLRI-2000-31237, QLRT-2000-0136, QLRI-2001-00015, and to the NIH through Grant 1R01HL67465-01. Partial support is also acknowledged to the‘‘Universidad San Pablo-CEU’’ through project 17/02.The authors thank Dr. A.C. Steven for useful discus-sions during the early stages of the work and for com-ments on the manuscript.

Appendix A. Selection of a blob for a series expansion

A.1. Space series expansion

We shall study here what is the maximum signal fre-quency that can be safely reconstructed by a seriesexpansion on a BCC grid using a particular blob.

A BCC grid can be represented (Herman, 1998) as theset of points

254 M. Unser et al. / Journal of Structural Biology 149 (2005) 243–255

GBCC ¼ rBCC ¼ BBCC

i

j

k

0B@1CA

8><>:¼ gffiffiffi

3p

1 1 �1

1 �1 1

�1 1 1

0B@1CA i

j

k

0B@1CA; i; j; k 2 Z

9>=>;;

where g is the distance between two consecutive sampleson the same axis. Let f ðrÞ; r 2 R3 be a three-dimensionalfunction and let f ðrBCCÞ represent its samples on the lat-tice GBCC. Then the Fourier transform F of f is formedby replicas of the Fourier transform of f at the points ofthe reciprocal lattice of GBCC. The reciprocal lattice of aBCC grid is a Face Centered Cubic grid (FCC) given by

GFCC ¼ rFCC ¼ BFCC

i

j

k

0B@1CA

8><>:¼

ffiffiffi3

p

g

1 1 0

1 0 1

0 1 1

0B@1CA i

j

k

0B@1CA; i; j; k 2 Z

9>=>;:

Note that BFCC ¼ B�TBCC. The smallest distance between

two replicas in the reciprocal grid isffiffiffi6

p=g.

The samples f are convolved with the blob functionto build an approximation of the function to be recon-structed. Blobs are radial functions that are compactlysupported and, therefore, infinitely extended in the Fou-rier domain. The Kaiser–Bessel window function (Kai-ser, 1966), which is an approximation (van De Villeet al., 2002) of the zeroth order spheroidal wave function(Slepian and Pollak, 1961), has the property of maxi-mally compacting the energy in Fourier space. For thisreason, they can be thought of as ‘‘practically band-limited.’’ Let us call xbmax the ‘‘effective’’ bandwidthof the blob function and xmax the bandwidth of thesignal f. Since the distance between two consecutivesamples in the reciprocal space is

ffiffiffi6

p=g then, to have

an ‘‘effective’’ alias-free sampling, it must hold thatxbmax <

ffiffiffi6

p=g � xmax.

In the particular case of using ‘‘standard blobs’’ (Ma-tej and Lewitt, 1996), g ¼

ffiffiffi2

pT s where Ts is the sampling

rate of the images expressed in A per pixel. For this blobwe select xbmax = 1/Ts since above this frequency, thesignal is attenuated more than 70 dB; therefore,xmax <

ffiffi3

p�1

T s. This means that we cannot recover signals

with details smaller than 1.37 Ts without aliasing.

A.2. Fourier series expansion

Following a similar reasoning, if we sample the Fou-rier space using blobs on a BCC grid, thenTmax <

ffiffi6

p

G � T bmax must hold. In the previous formulaTmax is the maximum size of the particle being studied,G is the distance between two consecutive blobs in the

same axis (this time in Fourier space), and Tbmax is themaximum size of the inverse Fourier transform of a bloblocated in the Fourier origin.

If we particularize for ‘‘standard blobs’’ thenG ¼

ffiffiffi2

pT sF, where TsF is the sampling rate in Fourier

space. (For simplicity we assume that the sampling rateis the same in all directions.) On the other hand,TsF = 1/Ta where Ta is the space available for the vol-ume, i.e., Ta = XdimTs where Ts is the sampling rate inthe image space in A per pixel and Xdim is the lengthin voxel units of the volume available to represent theparticle at hand. Under these conditions Tmax <T að

ffiffiffi3

p� 1Þ, or what is the same, the particle diameter

should not exceed the 74% of the available length inany of the directions.

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